The paper studies quadratic and Koszul duality for modules over positivelygraded categories. Typical examples are modules over a path algebra, which isgraded by the path length, of a not necessarily finite quiver with relations.We present a very general definition of quadratic and Koszul duality functorsbacked up by explicit examples. This generalises previous results in twosubstantial ways: We work in the setup of graded categories, i.e. we allowinfinitely many idempotents and also define a ``Koszul'' duality functor fornot necessarily Koszul categories. As an illustration of the techniques wereprove the Koszul duality of translation and Zuckerman functors for theclassical category O in a quite elementary and explicit way. As applications wepropose a definition of a "Koszul" dual category for integral blocks ofHarish-Chandra bimodules and for blocks outside the critical hyperplanes forthe Kac-Moody category O.
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